Further Analysis of an Outcome Set-Based Algorithm for Multiple-Objective Linear Programming

Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, Benson recently developed a finite, outer-approximation algorithm for generating the set of all efficient extreme points in the outcome set, rather than in the decision set, of problem (MOLP). In this article, we show that the Benson algorithm also generates the set of all weakly efficient points in the outcome set of problem (MOLP). As a result, the usefulness of the algorithm as a decision aid in multiple objective linear programming is further enhanced.     

Hybrid Approach for Solving Multiple-Objective Linear Programs in Outcome Space

Various difficulties arise in using decision set-based vector maximization methods to solve a multiple-objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have begun to develop tools for analyzing and solving problem (MOLP) in outcome space, rather than in decision space. In this article, we present and validate a new hybrid vector maximization approach for solving problem (MOLP) in outcome space. The approach systematically integrates a simplicial partitioning technique into an outer approximation procedure to yield an algorithm that generates the set of all efficient extreme points in the outcome set of problem (MOLP) in a finite number of iterations. Some key potential practical and computational advantages of the approach are indicated.     

A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program

Various computational difficulties arise in using decision set-based vector maximization methods to solve multiple objective linear programming problems. As a result, several researchers have begun to explore the possibility of solving these problems by examining subsets of their outcome sets, rather than of their decision sets. In this article, we present and validate a basic weight set decomposition approach for generating the set of all efficient extreme points in the outcome set of a multiple objective linear program. Based upon this approach, we then develop an algorithm, called the Weight Set Decomposition Algorithm, for generating this set. A sample problem is solved using this algorithm, and the main potential computational and practical advantages of the algorithm are indicated.     

Using Efficient Anchoring Points for Generating Search Directions in Interior Multiobjective Linear Programming

This paper modifies the affine-scaling primal algorithm to multiobjective linear programming (MOLP) problems. The modification is based on generating search directions in the form of projected gradients augmented by search directions pointing toward what we refer to as anchoring points. These anchoring points are located on the boundary of the feasible region and, together with the current, interior, iterate, define a cone in which we make the next step towards a solution of the MOLP problem. These anchoring points can be generated in more than one way. In this paper we present an approach that generates efficient anchoring points where the choice of termination solution available to the decision maker at each iteration consists of a set of efficient solutions. This set of efficient solutions is being updated during the iterative process so that only the most preferred solutions are retained for future considerations. Current MOLP algorithms are simplex-based and make their progress toward the optimal solution by following an exterior trajectory along the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an acceptable solution may prove less sensitive to problem size. We refer to the resulting class of MOLP algorithms that are based on the affine-scaling primal algorithm as affine-scaling interior multiobjective linear programming (ASIMOLP) algorithms.     

Approximating the nondominated set of an MOLP by approximately solving its dual problem

The geometric duality theory of Heyde and Löhne (2006) defines a dual to a multiple objective linear programme (MOLP). In objective space, the primal problem can be solved by Benson’s outer approximation method (Benson 1998a,b) while the dual problem can be solved by a dual variant of Benson’s algorithm (Ehrgott et al. 2007). Duality theory then assures that it is possible to find the (weakly) nondominated set of the primal MOLP by solving its dual. In this paper, we propose an algorithm to solve the dual MOLP approximately but within specified tolerance. This approximate solution set can be used to calculate an approximation of the weakly nondominated set of the primal. We show that this set is a weakly ε-nondominated set of the original primal MOLP and provide numerical evidence that this approach can be faster than solving the primal MOLP approximately.     

Parametric LP for sensitivity analysis of efficiency in MOLP problems

This paper describes sensitivity analysis in multiple objective linear programming (MOLP) with one of the criteria function coefficients parameterized. The parametric linear programming (LP) is used for analyzing a range set—the parameters set for which a given feasible solution is efficient for MOLP. The main theoretical result is a presentation of convexity of the range set. Moreover, an algorithm based on some of LP problems is presented for generating the range set.     

Branch-and-Bound Variant of an Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem

The paper presents a finite branch-and-bound variant of an outcome-based algorithm proposed by Benson and Lee for minimizing a lower-semicontinuous function over the efficient set of a bicriteria linear programming problem. Similarly to the Benson-Lee algorithm, we work primarily in the outcome space. Dissimilarly, instead of constructing a sequence of consecutive efficient edges in the outcome space, we use the idea of generating a refining sequence of partitions covering the at most two-dimensional efficient set in the outcome space. Computational experience is also presented.     

A modified method for constructing efficient solutions structure of MOLP

This paper deals with a recently proposed algorithm for obtaining all weak efficient and efficient solutions in a multi objective linear programming (MOLP) problem. The algorithm is based on solving some weighted sum problems, and presents an easy and clear solution structure. We first present an example to show that the algorithm may fail when at least one of these weighted sum problems has not a finite optimal solution. Then, the algorithm is modified to overcome this problem. The modified algorithm determines whether an efficient solution exists for a given MOLP and generates the solution set correctly (if exists) without any change in the complexity.     

An outcome space algorithm for optimization over the weakly efficient set of a multiple objective nonlinear programming problem

This article presents for the first time an algorithm specifically designed for globally minimizing a finite, convex function over the weakly efficient set of a multiple objective nonlinear programming problem (V1) that has both nonlinear objective functions and a convex, nonpolyhedral feasible region. The algorithm uses a branch and bound search in the outcome space of problem (V1), rather than in the decision space of the problem, to find a global optimal solution. Since the dimension of the outcome space is usually much smaller than the dimension of the decision space, often by one or more orders of magnitude, this approach can be expected to considerably shorten the search. In addition, the algorithm can be easily modified to obtain an approximate global optimal weakly efficient solution after a finite number of iterations. Furthermore, all of the subproblems that the algorithm must solve can be easily solved, since they are all convex programming problems. The key, and sometimes quite interesting, convergence properties of the algorithm are proven, and an example problem is solved.     

Determining maximal efficient faces in multiobjective linear programming problem

Finding an efficient or weakly efficient solution in a multiobjective linear programming (MOLP) problem is not a difficult task. The difficulty lies in finding all these solutions and representing their structures. Since there are many convenient approaches that obtain all of the (weakly) efficient extreme points and (weakly) efficient extreme rays in an MOLP, this paper develops an algorithm which effectively finds all of the (weakly) efficient maximal faces in an MOLP using all of the (weakly) efficient extreme points and extreme rays. The proposed algorithm avoids the degeneration problem, which is the major problem of the most of previous algorithms and gives an explicit structure for maximal efficient (weak efficient) faces. Consequently, it gives a convenient representation of efficient (weak efficient) set using maximal efficient (weak efficient) faces. The proposed algorithm is based on two facts. Firstly, the efficiency and weak efficiency property of a face is determined using a relative interior point of it. Secondly, the relative interior point is achieved using some affine independent points. Indeed, the affine independent property enable us to obtain an efficient relative interior point rapidly.     

Finding DEA-efficient hyperplanes using MOLP efficient faces

This paper suggests a method for finding efficient hyperplanes with variable returns to scale the technology in data envelopment analysis (DEA) by using the multiple objective linear programming (MOLP) structure. By presenting an MOLP problem for finding the gradient of efficient hyperplanes, We characterize the efficient faces. Thus, without finding the extreme efficient points of the MOLP problem and only by identifying the efficient faces of the MOLP problem, we characterize the efficient hyperplanes which make up the DEA efficient frontier. Finally, we provide an algorithm for finding the efficient supporting hyperplanes and efficient defining hyperplanes, which uses only one linear programming problem.     

Anchoring Points and Cones of Opportunities in Interior Multiobjective Linear Programming

This paper presents a modification of one variant of Karmarkar's interior-point linear programming algorithm to Multiobjective Linear Programming (MOLP) problems. We show that by taking the variant known as the affine-scaling primal algorithm, generating locally-relevant scaling coefficients and applying them to the projected gradients produced by it, we can define what we refer to as anchoring points that then define cones in which we search for an optimal solution through interaction with the decision maker. Currently existing MOLP algorithms are simplex-based and make their progress toward the optimal solution by following the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an optimal solution may prove less sensitive to problem size. We refer to the class of MOLP algorithms resulting from this variant as Affine-Scaling Interior Multiobjective Linear Programming (ASIMOLP) algorithms.     

Algorithms for nonlinear integer bicriterion problems

Two algorithms to solve the nonlinear bicriterion integer mathematical programming (BIMP) problem are presented. One is a noninteractive procedure that generates the entire efficient set, and the second one is an interactive procedure that determines the best compromise solution of the decision maker (DM). A Tchebycheff norm related approach is used for generating the efficient points for the BIMP problem. An application of the interactive procedure for a quality control problem is also presented.This research was supported by the National Science Foundation Grant No. ECS-82-12076 with the University of Oklahoma.     

Sensitivity analysis on the priority of the objective functions in lexicographic multiple objective linear programs

This paper deals with a class of multiple objective linear programs (MOLP) called lexicographic multiple objective linear programs (LMOLP). In this paper, by providing an efficient algorithm which employs the preceding computations as well, it is shown how we can solve the LMOLP problem if the priority of the objective functions is changed. In fact, the proposed algorithm is a kind of sensitivity analysis on the priority of the objective functions in the LMOLP problems.     

An MOLP based procedure for finding efficient units in DEA models

In this paper a multiple objective linear programming (MOLP) problem whose feasible region is the production possibility set with variable returns to scale is proposed. By solving this MOLP problem by multicriterion simplex method, the extreme efficient Pareto points can be obtained. Then the extreme efficient units in data envelopment analysis (DEA) with variable returns to scale, considering the specified theorems and conditions, can be obtained. Therefore, by solving the proposed MOLP problem, the non-dominant units in DEA can be found. Finally, a numerical example is provided.     

An Outcome Space Branch and Bound-Outer Approximation Algorithm for Convex Multiplicative Programming

This article presents a new global solution algorithm for Convex Multiplicative Programming called the Outcome Space Algorithm. To solve a given convex multiplicative program (P _D), the algorithm solves instead an equivalent quasiconcave minimization problem in the outcome space of the original problem. To help accomplish this, the algorithm uses branching, bounding and outer approximation by polytopes, all in the outcome space of problem (P _D). The algorithm economizes the computations that it requires by working in the outcome space, by avoiding the need to compute new vertices in the outer approximation process, and, except for one convex program per iteration, by requiring for its execution only linear programming techniques and simple algebra.     

Outcome Space Partition of the Weight Set in Multiobjective Linear Programming

Approaches for generating the set of efficient extreme points of the decision set of a multiple-objective linear program (P) that are based upon decompositions of the weight set W⁰ suffer from one of two special drawbacks. Either the required computations are redundant, or not all of the efficient extreme point set is found. This article shows that the weight set for problem (P) can be decomposed into a partition based upon the outcome set Y of the problem, where the elements of the partition are in one-to-one correspondence with the efficient extreme points of Y. As a result, the drawbacks of the decompositions of W⁰ based upon the decision set of problem (P) disappear. The article explains also how this new partition offers the potential to construct algorithms for solving large-scale applications of problem (P) in the outcome space, rather than in the decision space.     

Using a facility location algorithm to solve large set covering problems

Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncapacitated facility location problem (UFLP). In this paper, we use his algorithm to solve large instances of an important subset of the UFLP; the set covering problem (SCP). In addition, we present further empirical evidence that a heuristic algorithm developed by Vasko and Wilson for the SCP is capable of quickly generating good solutions to large SCP's.     

A Semidefinite Programming approach for solving Multiobjective Linear Programming

Several algorithms are available in the literature for finding the entire set of Pareto-optimal solutions of Multiobjective Linear Programmes (MOLPs). However, all of them are based on active-set methods (simplex-like approaches). We present a different method, based on a transformation of any MOLP into a unique lifted Semidefinite Program (SDP), the solutions of which encode the entire set of Pareto-optimal extreme point solutions of any MOLP. This SDP problem can be solved, among other algorithms, by interior point methods; thus unlike an active set-method, our method provides a new approach to find the set of Pareto-optimal solutions of MOLP.